Global quantities in algebraic quantum mechanics of infinite systems: Classical observables or parameters?

Abstract

Local observables of an infinite system are described by a quasilocal C*-algebra A. Traditionally classical observables are introduced as parameters that can be assigned to certain states on the local observables. As fundamental theories should be essentially free of parameters this can hardly be regarded as satisfactory. In the present paper classical observables are introduced explicitly as operators by a transitive system of imprimitivity of a kinematical group in the center of the W*-algebra π(A)″ belonging to a representation π of A. The states of the system are represented by linear functionals on π(A)″, which are dispersion-free on the center of π(A)″. For classical observables related to a kinematical group a W*-description—with algebra π(A)″ and its associated linear functionals that are dispersion-free on the center and normal on the noncentral part of π(A)″—is equivalent to a C*-description—with algebra A and a distinguished family of linear functionals on A, determined uniquely by π. The implications of these results on the interpretation of quantum mechanics of infinite systems are discussed.